A splitting method for incompressible flows with variable density based on a pressure Poisson equation

A new fractional time-stepping technique for solving incompressible flows with variable density is proposed. The main feature of this method is that, as opposed to other known algorithms, the pressure is determined by just solving one Poisson equation per time step, which greatly reduces the computational cost. The stability of the method is proved and the performance of the method is numerically illustrated.

[1]  R. Rannacher.  On chorin's projection method for the incompressible navier-stokes equations , 1992 .

[2]  R. Temam.  Une méthode d'approximation de la solution des équations de Navier-Stokes , 1968 .

[3]  P. Colella,et al.  A Conservative Adaptive Projection Method for the Variable Density Incompressible Navier-Stokes Equations , 1998 .

[4]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: II. Beyond SUPG , 1986 .

[5]  Jean-Luc Guermond,et al.  Some implementations of projection methods for Navier-Stokes equations , 1996 .

[6]  Jean-Luc Guermond,et al.  On the approximation of the unsteady Navier–Stokes equations by finite element projection methods , 1998, Numerische Mathematik.

[7]  T. Hughes,et al.  The Galerkin/least-squares method for advective-diffusive equations , 1988 .

[8]  Jean-Luc Guermond.  Un résultat de convergence d'ordre deux pour l'approximation des équations de Navier-Stokes par projection incrémentale , 1997 .

[9]  Noel Walkington,et al.  Convergence of the Discontinuous Galerkin Method for Discontinuous Solutions , 2004, SIAM J. Numer. Anal..

[10]  M. Minion,et al.  Accurate projection methods for the incompressible Navier—Stokes equations , 2001 .

[11]  Navier-Stokes equations,et al.  ON ERROR ESTIMATES OF PROJECTION METHODS FOR , 1992 .

[12]  Jean-Luc Guermond,et al.  Un résultat de convergence d'ordre deux en temps pour l'approximation des équations de Navier-Stokes par une technique de projection incrémentale , 1999 .

[13]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[14]  Jean-Luc Guermond,et al.  Entropy-based nonlinear viscosity for Fourier approximations of conservation laws , 2008 .

[15]  J. Guermond,et al.  Theory and practice of finite elements , 2004 .

[16]  J. Bell,et al.  A Second-Order Projection Method for Variable- Density Flows* , 1992 .

[17]  R. Temam.  Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (II) , 1969 .

[18]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[19]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multi-dimensional advective-diffusive systems , 1987 .

[20]  Jie Shen,et al.  An overview of projection methods for incompressible flows , 2006 .

[21]  Jie Shen,et al.  E-cient Chebyshev-Legendre Galerkin Methods for Elliptic Problems , 1996 .

[22]  Jie Shen,et al.  Gauge-Uzawa methods for incompressible flows with variable density , 2007, J. Comput. Phys..

[23]  P. Lions.  Mathematical topics in fluid mechanics , 1996 .

[24]  Jie Shen,et al.  On the error estimates for the rotational pressure-correction projection methods , 2003, Math. Comput..

[25]  Jean-Luc Guermond,et al.  Calculation of Incompressible Viscous Flows by an Unconditionally Stable Projection FEM , 1997 .

[26]  A. Chorin.  Numerical Solution of the Navier-Stokes Equations* , 1989 .

[27]  T. F. Russell,et al.  NUMERICAL METHODS FOR CONVECTION-DOMINATED DIFFUSION PROBLEMS BASED ON COMBINING THE METHOD OF CHARACTERISTICS WITH FINITE ELEMENT OR FINITE DIFFERENCE PROCEDURES* , 1982 .

[28]  Chun Liu,et al.  Convergence of Numerical Approximations of the Incompressible Navier-Stokes Equations with Variable Density and Viscosity , 2007, SIAM J. Numer. Anal..

[29]  Jean-Luc Guermond,et al.  Subgrid stabilized projection method for 2D unsteady flows at high Reynolds numbers , 2006 .

[30]  Claes Johnson,et al.  Finite element methods for linear hyperbolic problems , 1984 .

[31]  L. Quartapelle,et al.  A projection FEM for variable density incompressible flows , 2000 .

[32]  Abner J. Salgado,et al.  A fractional step method based on a pressure Poisson equation for incompressible flows with variable density , 2008 .

[33]  G. Tryggvason.  Numerical simulations of the Rayleigh-Taylor instability , 1988 .

[34]  Frans N. van de Vosse,et al.  An approximate projec-tion scheme for incompressible ow using spectral elements , 1996 .

[35]  Jean-Luc Guermond,et al.  Approximation of variable density incompressible flows by means of finite elements and finite volumes , 2001 .

[36]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.